Asymptotic EfficiEncy of oLs Under the Gauss-Markov assumptions, let b|
j denote estimators that solve equations of the form (5.19) and let b^j denote the OLS estimators. Then for j50, 1, 2, p, k, the OLS estimators have the smallest asymptotic variances: Avar!n1b^j2 bj2 #Avar!n1b|
j2 bj2.
Asymptotic Bias Asymptotic Confidence
Interval
Asymptotic Normality Asymptotic Properties Asymptotic Standard Error
Asymptotic t Statistics Asymptotic Variance Asymptotically Efficient Auxiliary Regression Consistency Inconsistency
Lagrange Multiplier (LM) Statistic
Large Sample Properties n-R-Squared Statistic Score Statistic
Problems
1 In the simple regression model under MLR.1 through MLR.4, we argued that the slope estimator, b^1, is consistent for b1. Using b^05y2 b^1x1, show that plim b^05 b0. [You need to use the consistency of b^1 and the law of large numbers, along with the fact that b05E1y2 2 b1E1x12.]
2 Suppose that the model
pctstck5 b01 b1funds1 b2risktol1u
satisfies the first four Gauss-Markov assumptions, where pctstck is the percentage of a worker’s pension invested in the stock market, funds is the number of mutual funds that the worker can
choose from, and risktol is some measure of risk tolerance (larger risktol means the person has a higher tolerance for risk). If funds and risktol are positively correlated, what is the inconsistency in b|
1, the slope coefficient in the simple regression of pctstck on funds?
3 The data set SMOKE contains information on smoking behavior and other variables for a random sample of single adults from the United States. The variable cigs is the (average) number of cigarettes smoked per day. Do you think cigs has a normal distribution in the U.S. adult population? Explain.
4 In the simple regression model (5.16), under the first four Gauss-Markov assumptions, we showed that estimators of the form (5.17) are consistent for the slope, b1. Given such an estimator, define an esti- mator of b0 by b|
05y2 b|
1x. Show that plim b|
05 b0.
5 The following histogram was created using the variable score in the data file ECONMATH. Thirty bins were used to create the histogram, and the height of each cell is the proportion of observations falling within the corresponding interval. The best-fitting normal distribution—that is, using the sample mean and sample standard deviation—has been superimposed on the histogram.
0 .02 .04 .06 .08 .1
proportion in cell
20 40 60 80 100
course score (in percentage form)
(i) If you use the normal distribution to estimate the probability that score exceeds 100, would the answer be zero? Why does your answer contradict the assumption of a normal distribution for score?
(ii) Explain what is happening in the left tail of the histogram. Does the normal distribution fit well in the left tail?
Computer Exercises
C1 Use the data in WAGE1 for this exercise.
(i) Estimate the equation
wage5 b01 b1educ1 b2exper1 b3tenure1u.
Save the residuals and plot a histogram.
(ii) Repeat part (i), but with log(wage) as the dependent variable.
(iii) Would you say that Assumption MLR.6 is closer to being satisfied for the level-level model or the log-level model?
C2 Use the data in GPA2 for this exercise.
(i) Using all 4,137 observations, estimate the equation
colgpa5 b01 b1hsperc1 b2sat1u and report the results in standard form.
(ii) Reestimate the equation in part (i), using the first 2,070 observations.
(iii) Find the ratio of the standard errors on hsperc from parts (i) and (ii). Compare this with the result from (5.10).
C3 In equation (4.42) of Chapter 4, using the data set BWGHT, compute the LM statistic for testing whether motheduc and fatheduc are jointly significant. In obtaining the residuals for the restricted model, be sure that the restricted model is estimated using only those observations for which all vari- ables in the unrestricted model are available (see Example 4.9).
C4 Several statistics are commonly used to detect nonnormality in underlying population distributions.
Here we will study one that measures the amount of skewness in a distribution. Recall that any nor- mally distributed random variable is symmetric about its mean; therefore, if we standardize a sym- metrically distributed random variable, say z51y2 my2/sy, where my5E1y2 and sy5sd1y2, then z has mean zero, variance one, and E1z32 50. Given a sample of data 5yi: i51, p, n6, we can stan- dardize yi in the sample by using zi5 1yi2 m^y2/s^y, where m^y is the sample mean and s^y is the sample standard deviation. (We ignore the fact that these are estimates based on the sample.) A sample statistic that measures skewness is n21gni51z3i, or where n is replaced with (n 21) as a degrees-of-freedom ad- justment. If y has a normal distribution in the population, the skewness measure in the sample for the standardized values should not differ significantly from zero.
(i) First use the data set 401KSUBS, keeping only observations with fsize51. Find the skewness measure for inc. Do the same for log(inc). Which variable has more skewness and therefore seems less likely to be normally distributed?
(ii) Next use BWGHT2. Find the skewness measures for bwght and log(bwght). What do you conclude?
(iii) Evaluate the following statement: “The logarithmic transformation always makes a positive variable look more normally distributed.”
(iv) If we are interested in the normality assumption in the context of regression, should we be evaluating the unconditional distributions of y and log(y)? Explain.
C5 Consider the analysis in Computer Exercise C11 in Chapter 4 using the data in HTV, where educ is the dependent variable in a regression.
(i) How many different values are taken on by educ in the sample? Does educ have a continuous distribution?
(ii) Plot a histogram of educ with a normal distribution overlay. Does the distribution of educ appear anything close to normal?
(iii) Which of the CLM assumptions seems clearly violated in the model educ5 b01 b1motheduc1 b2fatheduc1 b3abil1 b4abil21u?
How does this violation change the statistical inference procedures carried out in Computer Exercise C11 in Chapter 4?
C6 Use the data in ECONMATH to answer this question.
(i) Logically, what are the smallest and largest values that can be taken on by the variable score?
What are the smallest and largest values in the sample?
(ii) Consider the linear model
score5 b01 b1colgpa1 b2actmth1 b3acteng1u.
Why cannot Assumption MLR.6 hold for the error term u? What consequences does this have for using the usual t statistic to test H0: b350?
(iii) Estimate the model from part (ii) and obtain the t statistic and associated p-value for testing H0: b350. How would you defend your findings to someone who makes the following state- ment: “You cannot trust that p-value because clearly the error term in the equation cannot have a normal distribution.”